Remark: Family 374. Sharp one-third stability of optimal transport maps

For uniform source measure $\rho$ on a compact convex body with interior in dimension at least two, quadratic optimal transport maps satisfy $\Vert T_{\mu }-T_{\nu }{\Vert }_{L^2(\rho )}\le C{W}_2(\mu ,\nu {)}^{1/3}$ uniformly over targets in a fixed compact set. The exponent is sharp, even for three-atom targets, disproving Letrouit's conjectured square-root bound.

The authors are OpenAI. This companion does not verify the proof. The claim is not a theorem of the Enter series and not a premise of any registered sheet. It does not close O.1. O.1 stays NOT_CLOSED. It is not a de Sitter state and not the QdS external-leg scheme. No quantum coefficient is admitted. No compilation is asserted.